Click here to load reader

Outline Moebius Transformations - VISGRAF ... Moebius Transformations for Manipulation and Visualization of Spherical Panoramas • Current Research at VISGRAF Lab! • Collaboration

  • View
    4

  • Download
    0

Embed Size (px)

Text of Outline Moebius Transformations - VISGRAF ... Moebius Transformations for Manipulation and...

  • Moebius Transformations

    and

    Omnidirectional Images

    Luiz Velho

    IMPA

    Outline

    • Moebius Tranformations

    - Mathematical Fundamentals

    • Omnidirectional Images

    - Basic Concepts

    - 360 Panoramas

    • Applications

    - Wide Field of View

    Moebius Transformations

    Möbius Transformations • Complex Map

    !

    !

    • Definition:

    M : C 7! C

    M(z) = az + b

    cz + d

    (ad� bc) 6= 0

    with

    Anatomy of M • Decomposition into Sequence

    translation

    inversion

    scaling and rotation

    translation

    m1(z) = z + d c

    m2(z) = 1 z

    m3(z) = (bc�ad)

    c2 z

    m4(z) = z + a c

    m4 �m3 �m2 �m1(z)

    Fixing the Inversion • Point at Infinity

    !

    !

    !

    • Extended Complex Plane

    1 1 1 = 0

    1 0 = 1

    bC = C [ {1}

  • Riemann Sphere • Stereographic Projection

    N 0

    point at infinity

    ẑ = (✓,�) 7! z = cot(�/2)ei✓

    Complex Projective Space • Isomorphism

    !

    !

    !

    !

    • Geometry and Algebra

    z 7! w = M(z) in Ĉ

    ẑ 7! ŵ in ⌃ induces

    Riemann

    Sphere

    Complex
 Plane

    Properties of M

    • Projective Linear Group (Lie Group)

    !

    • Preservation of:

    - Circles (lines to circles)

    - Angles (conformal)

    - Symmetry (w.r.t. circles)

    PGL(2,C)

    Defining M • Images of 3 points (e.g)

    !

    • Ratios and Uniqueness

    !

    !

    • Normalization

    (a/b), (b/c), (c/d)

    az+b cz+d = M(z) =

    kaz+kb kcz+kd

    (ad� bc) = 1

    • Ratio of 2 complex numbers

    !

    !

    • Two Cases

    Homogeneous Coordinates

    z = �1 �2

    = [�1, �2] 6= [0, 0]

    �2 6= 0

    �2 = 0

    z = 1

    z = �1/�2

    Cross Ratio • The unique


    
 sending

    !

    !

    !

    • Theorem:
 If M maps 4 points
 then, the cross-ratio is invariant.

    z 7! w = M(z)

    q, r, s 7! q̃, r̃, s̃ (w � q̃)(r̃ � s̃) (w � s̃)(r̃ � q̃) = [w, q̃, r̃, s̃] = [z, q, r, s] =

    (z � q)(r � s) (z � s)(r � q)

    p, q, r, s 7! p̃, q̃, r̃, s̃

  • Orientation Properties • Maps Oriented Circles to Oriented Circles

    - s.t. Regions are mapped accordingly

    Fixed Points • Solution of

    !

    • M has at most two fixed points

    - except for Id.

    • For M Normalized

    z = M(z)

    ⇠± = (a�d)±

    p (a+d)2�4

    2c

    M - Classification • Fixed Point at Infinity :

    !

    !

    • Basic Types

    - Elliptic

    - Hyperbolic

    - Parabolic

    - Loxodromic

    c = 0

    M(z) = Az +B • Rotation

    !

    !

    !

    !

    • two fixed points

    Elliptic Transform

    z 7! ei↵z

    (0,1)

    Hyperbolic Transform • Scaling

    !

    !

    !

    !

    • two fixed points

    z 7! ⇢z

    (0,1)

    • Rotation and Scaling

    !

    !

    !

    !

    • two fixed points

    Loxodromic Transform

    z 7! ⇢ei↵z

    (combination of elliptic and hyperbolic)

  • • Translation

    !

    !

    !

    !

    • one fixed point at

    Parabolic Transform

    z 7! z + b

    1

    Omnidirectional 
 Images

    Basic Concepts

    • Plenoptic Function

    • Capturing Light Fields

    • 360 Panoramas

    • Parametrization and Projections •

    Plenoptic Function

    Complete description of Visual Information in a 3D environment

    I� = P (x, y, z, ✓,�, t)

    P : R3 ⇥ S2 ⇥ R 7! E

    Holographic Image

    6D Phase Space x,y,z

    t

    ✓,�

    Light Field

    • Structured Sampling of P A Slice of the Plenoptic Function

    - example: Camera

    x,y,z

    Ray Space

    Image

    x,y,z fixed

    Omnindirectional Image

    • Spherical Light Field = 360 degrees

    The Set of All Rays incident at a point (x,y,z)

    Representation of Choice

  • Capturing Point Light Fields

    • Omnidirectional Cameras

    - Catadioptric

    - Dioptric

    - Multi-Camera

    Catadioptric Cameras • Mirror-Based (parabolic or hyperbolic)

    camera mirror image

    Dioptric Cameras • Fish Eye Lenses

    camera lens image

    Multi-Camera Systems • Point Grey's Ladybug (6 Perspective Cameras)

    Cam 0 Cam 1

    Cam 2

    Cam 4

    Cam 3

    Cam 5

    Cam 0 Cam 1 Cam 2 Cam 3 Cam 4 Cam 5

    Practical Option

    Panoramic Surfaces

    • Data Representation Generalized Support for Visual Information

    - example: Cilindrical Panorama

    Parametrizations

    • Coordinate Systems

    Maps 2D Surface to Planar Domain

    - example: Cilindrical Mapping

    h7!

  • 360˚ Image Formats

    !

    • Parametrizations of the Sphere

    - Lat-Long

    - Cube Map

    - Azimuthal

    - Stereographic (*)

    Omnidirectional Panoramas

    Equirectangular Projection

    • Latitude-Longitude Mapping (e.g., Flickr)

    natural coordinate system distortion toward poles

    Most Convenient Format

    Cube Mapping

    • 6 Perspective Projections

    suitable for CG rendering

    Azimuthal Projection • Hemispherical Mapping

    Dome Master standard

    Applications

    to


    360 Cinema

    Exhibition • Viewing Scenarios

    conventional theater full dome

  • Field of View • Reference to Observer

    - 30 to 90 degrees

    Film Language

    • Conventional Cinema

    - HD Television

    - Theater Panavision

    • 360 Degrees Dome

    - Omnimax

    - Dome Master

    Conventional Cinema • Camera Moves

    Track Pan / Tilt ZoomTrack Zoom

    360 Camera • Camera Moves

    Track Pan / Tilt Zoom

    ?yes maybe

    Authoring Issues

    • Passive

    - Movies

    • Interactive

    - Google Street View

    • Immersive

    - AR Cinema

    Emerging Technologies

    OBS: Post-Production

    Moebius Transformations for Manipulation and Visualization of Spherical Panoramas

    • Current Research at VISGRAF Lab

    • Collaboration with

    - Leonardo Koller Sacht

    - Luis Penaranda

    360˚ Image Transforms

  • Math of Camera Moves • Omnidirectional Images and


    Moebius Transformations

    - Pan / Tilt ⇔ Elliptic Transform

    - Zoom ⇔ Hyperbolic Transform

    !

    - Perspective ⇔ Parabolic Transform ?

    Transformation Pipeline • Möbius Mapping

    input

    equirectangular

    image

    complex

    plane

    representation

    scaled

    complex

    image

    final

    equirectangular

    image

    S(i) S�1(i)M

    Example • Extreme Zoom

    Comparison • Alternative Projections

    input panorama

    equirectangular

    equi-rectangular

    projective mercator möbius

    Control of

    Perspective

    Current Work

    • Preserving Lines

    !

    • Perspective Control

  • Questions?

Search related